This sequence looks superficially like a simple continued-fraction expansion of some constant, but it's not. There are no numbers (I conjecture) ending with the digit 5. How did I arrive at my sequence? They are the magic multipliers (the final number on each line) in the following:

The initial number on each line is A135952(n). Note that if the magic multiplier is odd, we add 1 before dividing by 2; if it is even, we subtract 1 before dividing by 2. The number immediately after each equal sign is the prime p where A135952(n) dividescomposite Fibonacci(p). In my sequence of magic multipliers, the first occurrences of the positive integers are at indices: